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As the EU and the UK prepare for divorce, it is the £47.5bn question. How will Britain settle its budget affairs with Europe? And what will the EU do to fill the black hole if Britain’s contributions to the club stop in years to come?

The EU’s common budget has been the subject of epic battles, stretching back to Margaret Thatcher’s legendary defence of British interests at the 1984 Fontainebleau summit, where she secured a rebate that has kept down Britain’s net contributions to this day.

Following the UK’s vote to leave the EU, officials in London and Brussels are scrambling to find ways to engineer a break-up settlement between pound-wise Britain and the relatively spendthrift EU. It could easily turn messy.

Not only will both sides be trying to use the more than £9bn a year Britain sends Brussels to their advantage, but the final deal could be an important catalyst for a rethink on how to finance the EU budget.

Money is often one of the stickiest issues in divorces; this case is likely to be no exception.

Why is the budget important for Britain?The British public’s reservations about contributing to the EU budget were at the heart of the successful Leave campaign, whose election broadcasts and posters told voters that Brexit would allow Britain to stop giving £350m a week to Brussels and suggested the money could go to the National Health Service instead.

In fact, Britain’s net contributions are, at most, about half that figure, according to UK data — brought down not just by the rebate that Thatcher won at Fontainebleau but also by EU funds to the UK state and private sectors.

Since the referendum, leading members of the Leave campaign, such as Iain Duncan Smith, Britain’s former work and pensions secretary, have partly rowed back on the £350m commitment to the NHS. But Brexiters will still be watching impatiently for the cheques to Brussels to stop.

How will this affect the negotiations?UK contributions are hard-wired into the EU’s long-term budget, which totals €960bn and covers the years from 2014 to 2020. Britain’s commitment is set to run until several years after that, due to a time lag between money being allocated and actually being handed over.

A UK House of Commons report last week estimated that, under the pre-Brexit status quo, Britain would have made a net contribution of £65.7bn from 2014 to 2020, with £47.5bn of the total coming in the years 2016-2020.

The European Commission’s estimate for Britain’s net contributions is considerably lower, since it factors out payments to the UK private sector — such as farm subsidies or research funding — as well as the transfers to state bodies already deducted from the British numbers.

The long-term planning for the EU budget creates a practical headache for the exit talks, as the budget is a collection of delicate political compromises. Trying to extricate the UK from it could prompt a wider row about financing the EU. Changing the budget requires unanimous EU agreement — an even higher bar than the one needed to sign off Britain’s general divorce settlement with the bloc, which requires a qualified majority of member states to approve the terms.

There is also the tricky question of what should happen if the UK is still in the EU when the next multi-annual budget starts to be negotiated. Liam Fox, Conservative leadership candidate and former defence secretary, has said that he is keen for Britain to leave the EU no later than January 1 2019 to avoid being dragged into the next round of budget planning.

Could Britain simply refuse to keep contributing?This has been ruled out by David Cameron, the prime minister. The problem is what should be done if Britain wants to leave before the multi-annual budget runs out, as its contributions constitute a liability taken on by the British state.

Simply refusing to pay could damage the UK’s reputation in the eyes of its creditors. “Not paying would be breaking a legal framework,” one European Commission official told the Financial Times, suggesting that there could be repercussions for the UK’s credit rating.

The solution will need to be found in Britain’s exit talks.

The contributions are a stick for the EU side too. Should Britain drag out divorce talks and refuse to invoke Article 50 — the formal EU exit clause — it will have to keep on paying in.

So what’s the solution?EU officials hope Britain will honour its obligations as a parting “goodwill gesture”. The next prime minister may have other ideas. Another simpler outcome could be that the Brexit talks drag on so long that it ceases to be a material issue.

Who will fill the gap when Britain leaves?Some EU diplomats predict a game of nip and tuck — that the big net contributors will agree to give a little more while the big net recipients in eastern and southern Europe will agree to receive a little less.

Another, more radical, solution would be to expand the sources of revenue that go directly into the EU budget, bypassing national treasuries entirely. The main ones at present are sugar and customs duties and a share of VAT revenues.

Options that have been mooted by the European Parliament and others down the years include allocating revenues from the EU’s emissions trading scheme to the EU, enlarging its share of VAT receipts or even imposing new taxes on the financial sector.

EU officials see an opportunity, as the UK has been the staunchest opponent of further moves in this direction. The big question is to what extent other reluctant governments have been hiding behind the UK.